All Stats Formulas

Question 1

Standard deviation

Answer 1

√{ Σx² - n(x̄)² } / n

Question 2

Coded mean

Answer 2

{ Σ(x ± a) / n } ∓ a

Question 3

Coded standard deviation

Answer 3

√ { Σ(x ± a)²/ n } - { [Σ(x ± a) / n]² }

Question 4

Combined mean

Answer 4

[Σx + Σy] / [nₓ + nᵧ]

Question 5

Combined standard deviation

Answer 5

√ { [Σx² + Σy²] / [nₓ + nᵧ] } - { [Σx + Σy / nₓ + nᵧ]² }

Question 6

Applied mean

Answer 6

1/a{ Σ(ax ± b) / n } ∓ b

Question 7

Applied standard deviation

Answer 7

1/a √ { Σ(ax ± b)²/ n } - { [Σ(ax ± b) / n]² }

Question 8

Objects in a circle (without repeats)

Answer 8

(n -1)!

Question 9

Independent events

Answer 9

P(A & B) = P(A ∩ B) = P(A) x P(B)

Question 10

Conditional probability

Answer 10

P(A|B) = { P(A ∩ B) } / P(B)

Question 11

Outlier

Answer 11

1.5x greater the UQ or 1.5x lesser than LQ

Question 12

Skewness

Answer 12

skew refers to tail so positive skew has tail towards positive and negative skew has tail towards negative

Question 13

Mutually exclusive events

Answer 13

P(A ∩ B) = 0 , A and B can not occur together

Question 14

Probability summation

Answer 14

Σ P(X = x) = 1

Question 15

Expectation

Answer 15

E(X) = µ = ∑ xᵢPᵢ

Question 16

Variance of a random variable (2)

Answer 16

Var(X) = E(X - µ)² = ∑ (xᵢ - µ)² x P(xᵢ)

Var(X) = E(X²) - [E(X)]²

Question 17

Conditions/Notation of Binomial Distribution (5)

Answer 17

X ~ B (n, p)

1. Only 2 possible outcomes (discrete) usually called successes (p) or failures (q), [successes (p), outcome we are interested]
2. There are a fixed number of trials (n)
3. Each trial must be independent of the other trials
4. The probability of success (p) is fixed at each trial

Question 18

Binomial distribution (4)

Answer 18

P(X = x) = (n x)pˣ(1-p)ⁿ⁻ˣ = (n x)pˣqⁿ⁻ˣ

E(X) = np

Var(X) = npq

Sd(X) = √Var(X) = √npq

Question 19

Mode of Binomial Distribution

Answer 19

X value with the highest probability

Question 20

Conditions/Notation of Geometric Distribution

Answer 20

X ~ Geo (p)

1. Only 2 possible outcomes (discrete) usually called successes (p) or failures (q), [successes (p), outcome we are interested]
2. The repeated trials can be infinite
3. Each trial must be independent of the other trials
4. The probability of success (p) is fixed at each trial

Question 21

Geometric distribution (4)

Answer 21

P(X = x) = p(1 - p)ˣ⁻¹ = pqˣ⁻¹

E(X) = 1/p

Var(X) = q/p²

Sd(X) = √Var(X) = √q/p²

Question 22

Geometric Distributional Inequalities

Answer 22

Fewer than:
P(X > x) = (1 - p)ˣ = qˣ

At least:
P(X ≤ x) = 1 - (1 - p)ˣ = 1 - qˣ

Question 23

Mode of Geometric distribution

Answer 23

P(X = 1) has the greatest probability in all geometric distributions

Question 24

Standard normal distribution

Answer 24

X ~ N (µ, σ²) –> Z ~ N (0, 1)

Question 25

Standardising normal distribution

Answer 25

X ~ N (µ, σ²) where Z = [ x - µ ] / σ

Question 26

Normal approximation to binomial distribution

Answer 26

If X ~ B (n, p) where np > 5 and nq > 5 then

X ~ B (n, p) –> X ~ N (µ, σ²)